<p>Let <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> be the number of ways to place the integers <img alt="$1$" height="14" src="https://latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" width="11" /> through <img alt="$12$" height="12" src="https://latex.artofproblemsolving.com/e/d/f/edf074831eb5bc9e61d6d6e09f525a86e3068f6a.png" width="17" /> in the <img alt="$12$" height="12" src="https://latex.artofproblemsolving.com/e/d/f/edf074831eb5bc9e61d6d6e09f525a86e3068f6a.png" width="17" /> cells of a <img alt="$2 \times 6$" height="12" src="https://latex.artofproblemsolving.com/4/5/7/457a92fa44ed3ac774c75cab0d841dba0d96c6a5.png" width="40" /> grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by <img alt="$3.$" height="12" src="https://latex.artofproblemsolving.com/9/3/0/93032f1ab8b685d84a767a2fde0d9006a464b5e4.png" width="12" /> One way to do this is shown below. Find the number of positive integer divisors of <img alt="$N.$" height="12" src="https://latex.artofproblemsolving.com/0/b/4/0b4a6d41204181432504f1108ded79e4a49e0e03.png" width="19" /><img alt="\[\begin{array}{|c|c|c|c|c|c|} \hline \,1\, & \,3\, & \,5\, & \,7\, & \,9\, & 11 \\ \hline \,2\, & \,4\, & \,6\, & \,8\, & 10 & 12 \\ \hline \end{array}\]" height="42" src="https://latex.artofproblemsolving.com/f/d/b/fdba8f1061b07630343b8333d80a69ee33cbc5ca.png" width="197" /></p>

Question No: 399

Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$ $\begin{array}{|c|c|c|c|c|c|} \hline \,1\, & \,3\, & \,5\, & \,7\, & \,9\, & 11 \\ \hline \,2\, & \,4\, & \,6\, & \,8\, & 10 & 12 \\ \hline \end{array}$

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Tags: Algebra Combinatorics

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